Faithfulness of the 2-Braid Group via Zigzag Algebra in Type B

TL;DR

This paper proves the faithfulness of the 2-braid group in type B using zigzag algebra, providing an alternative proof to Rouquier's conjecture by relating bimodule categories over quiver algebras to Soergel bimodules.

Contribution

It establishes a new connection between type B zigzag algebra bimodules and type B Soergel bimodules, offering a novel proof of the 2-braid group faithfulness conjecture.

Findings

Category of bimodules over type B zigzag algebra is a quotient of type B Soergel bimodules.

Provides an alternative proof of Rouquier's conjecture on 2-braid group faithfulness.

Clarifies the algebraic structure linking zigzag algebras and braid group representations.

Abstract

We show that certain category of bimodules over a finite dimensional quiver algebra known as type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier's conjecture on the faithfulness of the 2-braid groups for type B.